A Note on the Acceleration of the Solar Wind
D.E. Scott, Ph.D. (EE)
Abstract: The solar wind (SW) has several enigmatic properties among which is the
unexplained high maximum velocity of the fast SW (~ 800 km/s). Additionally, after leaving
the Sun, this stream of charged particles accelerates – increases its velocity. Previously
proposed mechanisms have not fully explained how this increasing velocity can occur nor how
it can attain its high final value. A numerical analysis is performed on an assumed fast solar
wind velocity profile that quantitatively identifies the acceleration experienced by a proton
within that wind as a function of its radial distance. An electric-field strength required to
produce the assumed acceleration of such singly charged particles is also quantitatively
determined, presented and discussed. The application of Maxwell’s equations to a non-quasineutral plasma identifies the shape of a solely positive (+ion) charge density that will produce
this electric field in a spherical geometry such as the one in which the Sun finds itself.
1. Introduction
Charged particles (both electrons and positive ions) stream from the Sun in what has become
known as the solar ‘wind’. There are at least two different categories of this flow 1 , the slow solar
wind and the fast solar wind. These variations are described, in the accepted view, as follows:
1. The slow solar wind generally comes from lower solar latitudes. The slow solar wind of
the minimum type originates from above the more active regions on the Sun. The low
helium content in this type of solar wind indicates a greater release height. It is not clear
yet how the slow solar wind can be released at all. The slow wind of the maximum type
emerges from substantially larger areas distributed all over the Sun. It contains twice as
much helium as the minimum type. Typical values: Velocity = 250-400 km/s and Proton
density = 10.7 cm-3. The plasma of the slow solar wind resembles the composition of the
photosphere.
2. The fast solar wind apparently emanates from magnetically open coronal holes which are
representative of the inactive Sun, i.e. the ‘quiet’ Sun. Consistently, the ‘quiet’ type solar
wind is found in high-speed streams. Although the major coronal holes usually cover the
polar caps at latitudes beyond 40–60°, the solar wind emerging there over-expands
significantly and fills up all the heliosphere except for the 40° wide streamer belt close to
the heliomagnetic equator. Typical values: Velocity = 400-800 km/s and Proton density =
3.0 cm-3. Its composition resembles that of the corona.
Note that the fast solar wind has a maximum velocity roughly twice that of the slow wind, but its
density is less than a third of the slow wind. The problem referred to in 1., above, of “how the
slow wind can be released at all” is because, given the Sun’s gravity, its constituents theoretically
cannot achieve ‘escape velocity 2 ’ which is approximately 3 619 km/s.
2. Measurements of the Solar Wind
Quantitative data on the velocity profiles of both of these variants of the solar wind have been
obtained from a few in situ observations but predominantly from inferred measurements. It is well
known that the velocities of both types of wind increase (accelerate) with distance from the Sun.
Finding a consistent explanation of why, and how this acceleration occurs has been an elusive
goal.
Probes that have approached (or will approach) the Sun closely are:
 Luna 1 (1959), Earth to Moon mission. Confirmed the existence of the solar wind.


Mariner 2 (1962). Mission to Venus. No quantitative solar wind measurements taken.
Mariner 10 (1973). Flew near to Mercury which is 85 solar radii (SR) . No quantitative
solar wind measurements taken.
 Helios (1976), approached to within 0.29AU= 63.8 SR. Measured solar wind. Found 15
times as many micrometeorites at 0.29 AU than near Earth’s orbit.
 MESSENGER (2012). Mission to Mercury. No quantitative solar wind measurements
taken.
 Bepi-Columbo (2015?). Mission to Mercury. No quantitative solar wind measurements
are planned.
 Solar Probe Plus (2018?), 8.5 SR. Quantitative solar wind measurements planned, but not
closer than 8.5 SR.
All of these were (will be) in essentially equatorial orbits that do not reach higher solar latitudes
(the location of the fast solar wind).
In 1990, the Ulysses probe was launched to study the solar wind from high solar latitudes. Ulysses
did get into high solar latitudes (see figure 1), but only at great distances from the Sun – locations
where maximum solar wind velocity had already been attained (where the acceleration had
essentially stopped).
Indirect (inferred) ‘measurements’ include:
 Monitoring the activity of the auroras.
 Measuring the velocity of solar flares and CMEs.
 Radio scintillation studies.
Figure 1. Ulysses’ orbit.
3. Earlier Models of Solar Wind Velocities
Helio-astronomer Eugene Parker developed a model 4 that attempted to explain the acceleration of
the fast solar wind. It was based on the ideas of Sidney Chapman and involved hydrodynamics,
but not electro-dynamics. “Parker modeled the solar wind as a steady-state outflow. His model
assumes that the solar wind flux behaves like an ideal gas expanding isothermally into a vacuum.
The pressure contribution of the magnetic field is neglected. The solar wind flux velocity as a
function of radial distance is obtained from Euler's equation of motion and the equation of
continuity, assuming spherical symmetry and steady state (with all time derivatives equal to
zero).” Parker’s model did not achieve universal acceptance. One paper stated 5 , “However, the
acceleration of the fast wind is still not understood and cannot be fully explained by Parker's
theory.” Pressure implies collisions between particles, but the corona is essentially collisionless 6
plasma. Additionally, the Sun’s lower corona is not a good location to assume isothermal
2
reactions. Temperatures there vary from ~ 6,000 Kelvin to over 2 million K in a relatively short
distance.
A letter to the journal Nature 7 offered this comment:
“The solar wind is a supersonic outflow of coronal plasma into interplanetary space, and is
the agent that carries solar disturbances to the Earth. Direct measurements of the wind
speed over a range of distances—from the orbit of Mercury to beyond the outermost
planets and now over the solar poles – show that the acceleration is largely complete by
70 solar radii (SR). But, there are no direct measurements nearer the Sun with which to
constrain theoretical models of the acceleration. In principle, the speed of the solar wind in
the acceleration region can be inferred by indirect methods such as radio scattering, but
this is not straightforward as these data provide a measure of the wind properties
integrated along the lines of sight. Here we report radio-scattering measurements of the
speed of the south polar stream which have been corrected for this path integration, and
also for the potential bias due to the presence of plasma waves. Our results indicate that
the acceleration of the polar wind is almost complete by 10 SR much closer to the Sun
than had been expected. This suggests that the acceleration of the solar wind and the
heating of the solar corona occur in essentially the same region, and thus that the
underlying mechanisms may be strongly linked.” [Emphasis added.]
Then, in the late 1990s 8 , the Ultraviolet Coronal Spectrometer (UVCS) instrument on board
the SOHO spacecraft observed the acceleration region of the fast solar wind emanating from the
poles of the sun, and found that the wind accelerates much faster than can be accounted for by
thermodynamic expansion alone. Parker's model predicted that the wind should make the
transition to supersonic flow at an altitude of about 4 solar radii from the photosphere; but the
transition (or ‘sonic point’) now appears to be much lower, perhaps only 1 solar radius above the
photosphere, suggesting that some additional mechanism accelerates the solar wind away from
the sun. [Emphasis added.]
4. Present Modeling of the Solar Wind
More recently, in 2005, investigators at L’Observatoire de Paris have stated 9 :
“It has been more than four decades since the existence of the solar wind has been
confirmed by the measurements of the Mariner 2 spacecraft. However, the solar wind's
acceleration at supersonic speeds of about 700-800 km/s still remains unexplained.
Parker's theory, based on thermal conduction, results in a very low speed; this led most of
the scientists to look for an additional form of energy in order to explain this acceleration.
A team of astronomers working at LESIA 10 at the Paris Observatory has proposed an
alternative theory based on the role of electrons that are not in thermodynamic
equilibrium; these electrons would be the main driving force of the acceleration. This
approach explains, for the first time, the fast solar wind without any assumption of
additional energy.”
The LESIA group was the first to offer a model of SW acceleration that includes electrical
effects. This can be considered a breakthrough. The LESIA data, shown below in figure 2, is
clearly presented and that is the reason it is used as the basis for the numerical computations that
are presented here. However, the calculations being presented here do not support the LESIA
conclusion that negative charges (electrons) are involved in the acceleration process.
3
Figure 2. Solar wind speed profile derived by the LESIA group as a function of the heliocentric distance. The
upper (Kinetic model) curve shows the result of their model. The lower curve shows an example of Parker's
model.
For reference, the orbital distances of Mercury, Venus, and Earth are as follows:
Mercury
Venus
Earth
SR
84.97
158.74
220
AU
0.387
0.723
1
The LESIA data (Kinetic model curve above) exhibits a non-monotonic 11 inflection point which
gives rise to an acceleration maximum at very low (but greater than r = 1 SR) radial heliocentric
distance.
On March 8, 2013 NASA announced 12 they had discovered the source of the acceleration of the
solar wind. They used data from a older probe named Wind that has been at the Earth-Sun
Langrange point, L1, since 2004 (originally launched in 1994). Although this probe has never
approached the regions where solar wind acceleration occurs, the release confidently stated:
“The source of the heating in the solar wind is ion cyclotron waves. Ion cyclotron
waves are made of protons that circle in wavelike-rhythms around the sun's
magnetic field. According to a theory developed by Phil Isenberg (University of
New Hampshire) and expanded by Vitaly Galinsky and Valentin Shevchenko (UC
San Diego), ion cyclotron waves emanate from the sun; coursing through the solar
wind, they heat the gas to millions of degrees and accelerate its flow to millions of
miles per hour.”
Charged particles moving within a magnetic field will revolve (as they do in a cyclotron) at what
is called the gyroradius or Larmor radius. At higher energies such oscillations give rise to what is
termed magnetobremsstrahlung or synchrotron radiation 13 . This circular component of the
motion of electrically charged particles moving in a magnetic field is what gives rise to the helical
shape of Birkeland currents. The NASA release went on to say, “Chemical elements of the solar
4
wind such as hydrogen, helium, and heavier ions, blow at different speeds; they have different
temperatures; and, strangest of all, the temperatures change with direction. We have long
wondered why heavier elements in the solar wind move faster and have higher temperatures than
the lighter elements … This is completely counterintuitive.” It certainly is.
What should be expected from any collisional source of acceleration is that the affected particles
ought to be accelerated inversely as their masses. Something with three times the mass of a test
particle ought to be accelerated only one third as much as the test particle.
However, a possible cause of heavier ions moving faster than expected (almost as rapidly as, say,
monatomic hydrogen) is that they can become multiply ionized – the heavier the ion, the more the
opportunity to do so. But NASA does not mention any electric-field forces which might help to
resolve the ‘counterintuitivity’ of their observations. They suggest that plasma waves have the
ability to selectively accelerate massive particles more strongly than less massive particles. Their
description of the exact process remains somewhat superficial, however.
Nevertheless, NASA has now put forth the idea that ‘ion cyclotron waves’ produce the
acceleration of the solar wind despite the fact that three years ago, in a 2010 paper, 14 D.A.
Roberts stated explicitly that “hydromagnetic waves, whether turbulent or not, cannot produce the
acceleration of the fast solar wind and the related heating of the open solar corona. … In
particular, turbulence does not play a strong role in the corona as shown both by observations of
coronal striations and other features, … We consider possible ‘ways out’ of the arguments
presented, and suggest that in the absence of wave or turbulent heating and acceleration, the
chromosphere and transition region become the natural source, if yet unproven, of open
coronal energization through the production of nonthermal particle distributions.” [Emphasis
added.]
5. A Numerical Analysis of the LESIA Solar Wind Profile
Although figure 2, above, does not represent actual observed data, it is the velocity profile
generated by the LESIA group’s model. We accept it as being a reasonable approximation of real
data and use it to determine a quantitative account of the acceleration experienced by the ions in
the fast solar wind. In the work presented here, the heliocentric distance is denoted as either r or
x. The LESIA figure shows velocity as a function of distance, not time. Therefore, the slope of
that curve cannot be used directly in determining the acceleration which is a (t )  dv , not
dt
dv . If the upper curve in the figure is sampled at each of the distances plotted logarithmically
dx
along the x-axis, we obtain the first two columns (A and B) in Table 1. The remaining columns
are defined below the table.
Table 1 Numerical Analysis of the Velocity of the Solar Wind Presented in LESIA’s Kinetic Model (Figure 2)
A
B
X
(SR)
Vel
(km/s)
C
Delt
x
(SR)
D
E
Delt x
(km)
x (km)
F
x (m)
G
t (This
segment)(s)
H
t
(Total)
(s)
I
t
(To)(days)
J
Del
Vel
(km/s)
K
L
Acc
(km/s^2)
Acc
(m/s^2)
M
E (v/m)
1
10
1
681342
681342
6.81E+08
68134
68134
0.789
10
1.47E-04
0.147
1.47E-09
2
170
1
681342
1.36E+06
1.36E+09
4008
72142
0.835
160
3.99E-02
39.92
3.99E-07
3
315
1
681342
2.04E+06
2.04E+09
2163
74305
0.860
145
6.70E-02
67.04
6.70E-07
4
410
1
681342
2.73E+06
2.73E+09
1662
75967
0.879
95
5.72E-02
57.17
5.71E-07
5
470
1
681342
3.41E+06
3.41E+09
1450
77417
0.896
60
4.14E-02
41.39
4.13E-07
6
520
1
681342
4.09E+06
4.09E+09
1310
78727
0.911
50
3.82E-02
38.16
3.81E-07
7
550
1
681342
4.77E+06
4.77E+09
1239
79966
0.926
30
2.42E-02
24.22
2.42E-07
5
8
580
1
681342
5.45E+06
5.45E+09
1175
81140
0.939
30
2.55E-02
25.54
2.55E-07
9
595
1
681342
6.13E+06
6.13E+09
1145
82285
0.952
15
1.31E-02
13.10
1.31E-07
10
610
1
681342
6.81E+06
6.81E+09
1117
83402
0.965
15
1.34E-02
13.43
1.34E-07
20
675
10
6813420
1.36E+07
1.36E+10
10094
93496
1.082
65
6.44E-03
6.44
6.43E-08
30
690
10
6813420
2.04E+07
2.04E+10
9875
103371
1.196
15
1.52E-03
1.52
1.52E-08
40
698
10
6813420
2.73E+07
2.73E+10
9761
113132
1.309
8
8.20E-04
0.82
8.19E-09
50
705
10
6813420
3.41E+07
3.41E+10
9664
122797
1.421
7
7.24E-04
0.72
7.23E-09
60
710
10
6813420
4.09E+07
4.09E+10
9596
132393
1.532
5
5.21E-04
0.52
5.20E-09
70
712
10
6813420
4.77E+07
4.77E+10
9569
141962
1.643
2
2.09E-04
0.21
2.09E-09
80
715
10
6813420
5.45E+07
5.45E+10
9529
151492
1.753
3
3.15E-04
0.31
3.14E-09
90
718
10
6813420
6.13E+07
6.13E+10
9489
160981
1.863
3
3.16E-04
0.32
3.16E-09
100
720
10
6813420
6.81E+07
6.81E+10
9463
170444
1.973
2
2.11E-04
0.21
2.11E-09
200
725
100
68134200
1.36E+08
1.36E+11
93978
264422
3.060
5
5.32E-05
0.05
5.31E-10
Definition of the columns in Table 1:
C = Distance increment between points in solar radii (SR).
D = Distance increment between points (km).
E = Total radial distance measured from the Sun’s center (km).
F = Total radial distance measured from the Sun’s center (m).
G = Time taken by a particle to go from the previous point to this point, Δt = Δx/v = km/(km/sec) = sec = Dn/Bn.
H = Total time (sec) taken for the particle to get to this point.
I = Total time (days) taken for the particle to get to this point.
J = Gain in velocity from last point to this point = Bn- Bn-1 = Δv (in km/s).
K = Acceleration = Δv/Δt = Jn/Gn (in km/s2).
L = Acceleration = Δv/Δt = Jn/Gn (in m/s2).
M = Electric field strength under the assumption that f  eE  m p a . Mn = E = mp a/e = (1.6x10-27/1.60x10-19)Ln.
Plotting the first three points in column L, acceleration in m/sec2, we see an essentially linear
relationship of acceleration to distance at low altitudes above the solar surface:
First Linear Range
80.000
70.000
Acc (m /s^ 2)
60.000
50.000
y
40.000
Linear (y)
y = 33.445x - 31.188
R2 = 0.9882
30.000
20.000
10.000
0.000
0
0.5
1
1.5
2
2.5
3
3.5
Figure 3. Acceleration as
a function of radial
distance from the
photospheric surface of
the Sun up to 3 SR.
Solar Radii
The plot in figure 3 is monotonic and (almost) linear with a high value of R2, the coefficient of
determination. For greater radial distances, r > 3 SR, the acceleration plot is monotonically
decreasing as is shown in figure 4.
6
Accel. for 3<SR<100
120.000
Acceleration (m/sec/sec)
100.000
80.000
y
60.000
Power (y)
40.000
y = 739.98x -1.7799
R2 = 0.9835
20.000
0.000
0
20
40
60
80
100
120
Distance (SR)
Figure 4. Acceleration of the fast solar wind for distance, 3SR < r < 100 SR.
From this data, the acceleration experienced by a solar wind particle may be described 15 as:
a(r )  33.445r  31.188 for 0 < r < 3 SR
(1)
and
a(r )  739.98r 1.7799 for r > 3 SR
(2)
We assume the LESIA data is an accurate description of fast SW velocity but our use of it is, of
course, subject to any errors in our sampling process or subsequent computations. Subject to these
caveats, we will assume expressions (1) and (2) represent typical approximate data about the
acceleration of the fast solar wind. We can thus assume (using f = ma) that we now know the
applied accelerating force as a function of radial distance, f(r). However, these expressions do not
suggest a mechanism for achieving this force or acceleration.
6. An Electrical Mechanism
When seeking an effective mechanism for accelerating a charged particle such as a proton or
other positively charged ion, an obvious idea would be to investigate the effects of the presence of
an electric field. Although it is now becoming accepted that low valued E-fields can exist within
plasma, such a possibility has been a historical anathema in astrophysics. This author has written
about this as follows 16 :
Some early plasma researchers and many modern astrophysicists believe that plasmas are
perfectly conductive and so will ‘freeze’ magnetic fields within them. The typical plot of plasma
voltage vs. current demonstrates that this cannot happen. Every point on the plot (except the
origin) has a non-zero voltage coordinate. In other words, it requires a small electric field within
a plasma (no matter what mode it is in) to maintain it. No matter in which mode it is operating,
plasma is not an ideal conductor.
Another way of realizing this is to examine the resistivity of plasma. Consider the typical VI plot
shown in figure 5. The static resistivity of a plasma at any V-I operating point is equal to the slope of a
straight line drawn from the origin of the volt/ampere plot out to that point. This means that, in every
possible mode in which a plasma can operate, it has, in all of them, a non-zero static resistivity. A
non-zero-strength E-field is necessary to produce the current density. For example, the static
resistivity of a plasma in the high current end of the dark mode can be fairly large. In the arc mode,
static resistivity is typically extremely low – but not zero. The highest conductivity (lowest static
resistivity) plasmas are those operating in the arc mode. But even in that mode, any current density, no
matter how small, requires a non-zero-valued electric field. No plasma is an ‘ideal conductor.’
7
Many astrophysicists assume that Debye shielding is 100.000% effective in plasma at all scales.
However there are some exceptions to that generalization. Goedbloed 17 and Poedts in their text
on MHD (2004) stated: “By definition, plasmas are an interactive mix of charged particles,
neutrals, and fields that exhibits collective effects. In plasmas, charged particles are subject to
long-range, collective Coulomb interactions with many distant encounters. Although the
electrostatic force drops with distance (~1/r2), the combined effect of all charged particles might
not decay because the interacting volume increases as r3.”
VOLTAGE (V) or E-Field (V/m)
Dark Current Mode
Glow Mode
Townsend
Discharge
Arc Mode
E
D
C
Glow-to-arc
transition
H
F
Current
Saturation
F
Normal glow
I
G
Abnormal
glow
K
B
J
A
CURRENT (A) or CURRENT DENSITY (A/m2)
Figure 5. A typical static, plasma discharge, volt-ampere plot. This is representational of both the overall
terminal VI plot of a laboratory discharge and the point description of E-field vs. current density within the
plasma.
Under the influence of an electric field, the force of acceleration, fa, on an electronic charge, e, is
f a  e E  ma , so the electric field required to produce an acceleration, a, of a proton is:
mp
E (r ) 
a (r ) .
(3)
e
The E-field values required by equation 3 are tabulated in column M in Table 1. We propose that
mechanism as being the accelerating force for positive ions in the solar wind.
Note the extremely low numerical values in that column. The maximum value of required electric
field strength occurs at approximately 3SR and is ~0.7 μV/m. This is because of the extreme
effectiveness of the electric force in accelerating charged particles. These values are so low that
heavier (more massive) ions would still be accelerated by what would be considered surprisingly
low E-field strengths. And as noted earlier, when heavier elements are completely ionized (as
most are in the corona) they carry multiple positive charges and thus experience higher
accelerating forces than do single protons. This does not explain NASA’s observation that, “We
have long wondered why heavier elements in the solar wind move faster and have higher
temperatures than the lighter elements … This is completely counterintuitive.” But, the
acceleration of an ionized atom varies as the charge to mass ratio, q/m. Heavier ions will not be as
strongly accelerated as a lone proton, but they will be much more rapidly accelerated by an
8
electric field than they would be by any non-electrical force which would produce an acceleration
proportional only to 1/m.
An electric field that exists because of the presence of an electric charge distribution, ρ(r), obeys
Maxwell’s equation, Div E = ρ/ε. In the spherical geometry of the Sun’s surroundings 18,19 , this is
E  (r , ,  )
1  2
1 
E sin    1
div E  2
r Er 

given by:
r sin  
r sin  

r r
(4)


where ε is the permittivity of the plasma medium.
In the case where there is no azimuthal nor altitudinal variation, (4) becomes
1 d 2
 (r )
(5)

r E (r )  
2
r dr

Because of inherent imprecision in present evaluations of the permittivity of the plasma near the
Sun, we must leave ε as an unevaluated multiplicative constant. The permittivity value may be a
variable as well. Therefore, we can determine only the functional shape of ρ(r) 20 from (2), (3) and
(5) above, not the numerical value of its density. This results in:
62
 (r )  100 
for 1< r < 3 SR
(6)
x
and
 (r )  1675 x 2.7799 for 3 < r SR
(7)
These expressions describe a positive charge density that approaches zero asymptotically. See
figure 6.
R.R. Grail 21 stated that “Our results indicate that the acceleration of the polar wind is almost
complete by 10 SR much closer to the Sun than had been expected.” The present work agrees
well with that comment.
Charge Density
90
80
Arbitrary Values
70
60
50
40
30
20
10
0
0
5
10
15
20
25
Radial Distance, r (SR)
Figure 6. Charge Density as a function of radial distance from the Sun’s center. Vertical axis values are
arbitrary.
We conclude from the above discussion that the charge density shown in figure 6 and enumerated
in expressions (6) and (7) creates an electric field (which is the force per unit charge on the
particles in the solar wind). This force produces the acceleration on a proton shown in figures 3
9
and 4 and as described by expressions (1) and (2). This, in turn, produces the velocity profile
predicted by LESIA shown in figure 2.
It is important to note that the required charge density as shown in figure 6 is everywhere greater
than zero. Thus, only positive charges (+ions) are involved in the acceleration mechanism. No
participation of electrons in the required charge distribution is indicated.
7. An Extension of the Electronic Sun Hypothesis
It is not the purpose of this work to present a description of the Electronic Sun (ES)
hypothesis16,18 . However, it should be pointed out that the solar wind acceleration mechanism
described here is an obvious consequence of that model. The ES hypothesis states that outward
bound positive ions and protons (that will become constituents of the solar wind) rise up from the
photosphere, accelerate through a plasma double layer (DL) and collide with neutrals, other
atoms, and ions in the lower corona. Their radially directed (kinetic energy) velocity is thereby
brought almost to a standstill. Electrons that were associated with these ions drift downward, back
out of the lower corona, and serve to maintain the DL as per the Langmuir requirement 22 . These
electrons tend to fill the ‘electron trap’ that is formed by the relatively high-voltage photospheric
plasma granule cells. This is a probable cause of the relatively short life-times of those cells
(usually measured in hours or days).
The ionic kinetic energy transformed into thermal activity at the base of the corona produces the
greater than 2 million Kelvin temperatures measured there by spectroscopic observation. It also
results in the almost complete ionization of the coronal plasma. But then, what happens to those
positively charged ions? They are still there. These now relatively quiescent positive charges
constitute the ‘excess’ positive charge density region (over and above the quasi-neutral charge
densities of the background plasma) referred to in the above discussion and shown in figure 6.
8. Stability of the ‘Excess’ Positive Charge Density
A question that might be asked by
critics of this proposed mechanism for
producing the required electric field
strength is, “What maintains the
positive charge density in the shape
that it must maintain (shown in figure
6)?” To maintain a distribution of
matter that is more concentrated than it
would normally be in quiescent
conditions, there must be a power
input to it. A case-in-point is the
steady state large standing-wave that
can form at the bottom of certain water
slides. In figure 7, the wave would
collapse if the strong flow coming in
from the left were to stop. The height
Figure 7 Continuously stable standing-wave maintained by
of the water wave is higher than what the flow coming in from the left.
would be expected in a more slowly
moving river. A continuous flow of ions maintains the ‘excess’ charge density in figure 6.
10
9. Summary of Results
The purpose of this paper is to present a numerical analysis technique and the application of
Maxwell’s equations to a sample solar wind velocity profile that was generated by a model9
developed by the LESIA group at the Observatoire de Paris. It is not real solar wind speed data
but, it is typical of solar wind velocities described in other publications 23 . As more accurate, real
velocity data become available in the future, the numerical technique described here can easily be
re-applied to it. The results of this present sample analysis suggest the following conclusions:
 Including electric-field forces on the charged particles of the solar wind results in a model
wherein a distribution of excess positive ions in the lower corona accelerates solar wind
+ions to speeds of 700–800 km/s in the collisionless coronal plasma. Earlier models, such
as Parker’s that utilize only phenomena such as plasma waves, thermal turbulence, and
pressure gradients, have been unable to produce these levels of acceleration and velocity.
Why these mechanisms are still proposed in an acknowledged collisionless plasma is
unclear.
 A proton (a positive single electronic charge) in a collisionless environment can be
accelerated from rest to a velocity of 730 km/s by giving it 4.46 x 10-16 Joules of kinetic
energy. This is equivalent to ~2780 eV (allowing it to fall through a voltage drop of 2780
V). A piece-wise linear integration of the E-field values in column M (Table 1) yields a
value of over 3000 V. This total voltage is achieved, over large distances, by E-field
magnitudes that do not exceed one microvolt per meter (< 1 mV/km).
 Heavier (multiply ionized) particles can achieve accelerations almost as great as single
protons under the influence of such an E-field. Selective acceleration of heavy ions
however remains unexplained.
 Applying Maxwell’s equations in their proper spherical coordinate form is shown to be
essential in this geometry. Using rectangular (Cartesian) coordinates leads to a false
requirement that includes negative charges – electrons – in order to explain the decrease in
the wind’s acceleration beyond ~3 SR. But, electrons are not involved in the accelerating
mechanism derived here. Only a distribution of positive charges (protons and +ions) is
required as shown in figure 6 and enumerated in expressions 6 and 7.
 Raising the question of ‘escape velocity’ is inappropriate in a system of electric plasma
which is subject, over a great distance, to electrical forces that are much stronger than
collisional forces. (See 1. in the Introduction section of this paper.) In any event a ballistic
(explosive) acceleration process that would require attainment of an ‘escape velocity’ does
not take place. An electric–field provides a continuous, smooth acceleration.
 An important value that remains unknown is the value of the permittivity of the coronal
plasma which, in turn, would yield a numerical evaluation of the peak magnitude of the
required charge distribution. Without it, only the general shape of this distribution can be
obtained as has been done here.
 NASA’s Solar Probe Plus, now scheduled for a 2018 launch, will approach the Sun to
within a distance of 8.5 SR (0.034 AU = 5.9 million km). Even if the model presented
here is substantially correct, the predicted radial electric field (~0.2μV/m at that location)
will probably not be measurable even this close-in. It seems all or at least most of the
electrical ‘action’ occurs closer to the body of the Sun itself.
10. Closing Remark
It is surprising that it has taken so long for the electric-field to be considered as a possible causal
force in astrophysics. It was some sixty years ago that Hannes Alfvén wrote 24 ,
11
“Certainly we have seen plenty of evidence of electrical phenonena out in space. Within the
last few decades we have discovered several important electrical effects in the heavens:
strong stellar magnetic fields such as could only be caused by large electric currents, radio
waves emanating from the Sun and from many star systems, and the energetic cosmic rays,
which are electrically charged particles accelerated to tremendous speeds.” [Emphasis
added.]
The electrical mechanism presented here offers a uniquely simple and consistent explanation of
what, for decades, has been an enigma in helio-astronomy – discovering and quantifying the
process that causes the observed acceleration of the fast solar wind.
12
References and Endnotes
1
Solar Wind: Global Properties Encyclopedia Of Astronomy And Astrophysics.
Available: www.cfa.harvard.edu/~scranmer/ITC/eaaa_solar_wind_schwenn.pdf
2
A rocket moving out of a gravity well does not actually need to attain escape velocity to do so, but could achieve the
same result at any speed with a suitable mode of propulsion and sufficient fuel. Escape velocity only applies
to ballistic trajectories.
3
Solar Wind, NASA MSFC, Available: http://solarscience.msfc.nasa.gov/SolarWind.shtml
4
Parker, Solar wind model. http://demonstrations.wolfram.com/TheSolarWind/
5
G.W. Pneuman and R. A. Kopp (1971). Gas-magnetic field interactions in the solar corona, Solar Physics 18:
258. doi:10.1007/BF00145940
6
Collisionless plasma is plasma in which particles interact through the mutually induced space-charge field, and
collisions are assumed to be negligible. See: http://encyclopedia2.thefreedictionary.com/collisionless+plasma
Also see Plasma Physics of the Local Cosmos (2004), National Academies Press. Available:
http://www.nap.edu/openbook.php?record_id=10993&page=1
7
Grail, R.R., Rapid acceleration of the polar solar wind, Nature 379, 429 - 432 (01 February 1996);
doi:10.1038/379429a0
8
Available: http:// http://www.cfa.harvard.edu/uvcs/
9
Zouganelis, I., Explaining the acceleration of the fast solar wind
Available: http://www.obspm.fr/actual/nouvelle/jun05/solarw.en.shtml
10
Laboratoire d'études spatiales et d'instrumentation en astrophysique.
11
Zougalelis, I. et al., A transonic collisionless model of the solar wind, The Astrophysical Journal (May 2004) .
12
Solar Wind Energy Source Discovered, NASA Science News 3/8/2013.
Available: http://science.nasa.gov/science-news/science-at-nasa/2013/08mar_solarwind/
13
Peratt, A. L., Physics of the Plasma Universe, Springer-Verlag 1992, p. 197. Available:
http://link.springer.com/book/10.1007/978-1-4612-2780-9//page/1
14
Roberts, D.A., Demonstrations That The Solar Wind Is Not Accelerated By Waves Or Turbulence,
ApJ 711 1044 doi:10.1088/0004-637X/711/2/1044 Available: http://iopscience.iop.org/0004-637X/711/2/1044.
15
By least-squares fit with R2 = 0.9835.
16
Scott, D.E., The Electric Sky, Mikamar Pub., Portland, OR, 2006.
17
Goedbloed, H. and Poedts, S., Principles of Magnetohydrodynamics, Cambridge University Press, 2004.
18
Spherical coordinates Available: http://www4.wittenberg.edu/maxwell/CoordinateSystemReview.pdf
19
Scott, D.E., On the Sun’s Electric Field, Available: http://electric-cosmos.org/SunsEfield92210.pdf
20
Or consider that the result is simply a plot of ρ/ε.
21
Grail, R.R. op cit.
22
Levine, J.S. and Crawford, F.W., A Fluid Description of Plasma Double Layers, NASA Technical Report, SU-IPR
Report No. 787, p. 3.
May 1979 Available: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19790018771_1979018771.pdf
23
Available: http://umtof.umd.edu/pm/crn/CRN_1913.GIF
http://www.ann-geophys.net/20/1265/2002/angeo-20-1265-2002.pdf
http://hypertextbook.com/facts/2004/RomanOsatinski.shtml
http://web.mit.edu/afs/athena/org/s/space/www/voyager/voyager_data/voyager_data.html
24
H. Alfvén, The New Astronomy, Chapter 2, Section III, p. 74-79. Available:
http://www.plasma-universe.com/Texts:Electricity_in_Space_by_Hannes_Alfv%C3%A9n
13